Inter railing questions

Can you just step on a train anywhere in Europe without paying? I know it does not include Eurostar but am I right in thinking it pays for you to get from your local station to London?

We (2 sons and me) are planning London to Paris, Paris to Switzerland, Switzerland to Italy, Italy to Germany, France then home over a week or so in August. Looks to me that it will be cheaper for us to buy an inter railing ticket rather than so many different ones!

I believe for certain trains (specifically high-speed ones in Germany) you have to pay a zuschlag (supplement), and you have to pay for the bed in an overnight train. I also thought you could not use them for domestic travel, but i am willing to be corrected.